Leadership, Followership, and Beliefs About the World: Theory and Experiment

Transcription

1 Leadership, Followership, and Beliefs About the World: Theory and Experiment Eric S. Dickson Assistant Professor Department of Politics and Center for Experimental Social Science NewYorkUniversity January 13, 2008 Behavioral literature in political science and psychology suggests that factual beliefs about the world often vary across different social groups, including by partisan affiliation. This paper explores potential microfoundations for this regularity by analyzing interactions between a group leader and her followers. A game-theoretic framework is developed, in which a leader with private information about the state of the world sends a message about it to her followers. In the model, the leader is best off when followers successfully coordinate their actions. Because followers have preferences over coordination outcomes that are aligned in some states of the world, but not in others, leaders sometimes have incentives to misrepresent the state of the world in order to make coordination more likely. Two novel refinements, Leadership-Correlated Equilibrium and Belief-Based Followership Equilibrium, explicate distinct mechanisms through which a leader s messages about the world could potentially coordinate the actions of fully-rational followers. The intuitions behind these refinements are then tested in a laboratory experiment. Results from the experimental games suggest that leaders do frequently misrepresent the state of the world to followers when it is in their interests to do so; that leaders strategically-chosen messages about the state of the world are highly effective in coordinating group members actions; and that leaders messages strongly influence followers beliefs about the state of the world even when Bayesian followers would not find the messages to be credible as statements of fact. Followers appear not fully to account for leaders strategic incentives to misrepresent the state of the world in forming their posterior beliefs, even though the experimental elicitation mechanism offers substantial monetary rewards for accuracy. This finding suggests behavioral foundations for the empirical regularity that members of different social groups may have different factual beliefs about the world. Very preliminary draft; apologies for typos or infelicitous passages. eric.dickson@nyu.edu Comments and suggestions very welcome:

2 1 Introduction Leadership: the art of getting someone else to do something you want done because he wants to do it. -Dwight Eisenhower Leadership is among the most prominent themes in contemporary political discourse yet it is badly undertheorized within contemporary political science. Dwight Eisenhower s implicit claim that good leaders are effective at least in part because they are able to shape others preferences is both intuitively compelling and widespread in the public sphere. Yet, from a theoretical perspective, it is far from clear through what causal mechanism political or other group leaders might be expected to exert such influence. It is both natural and useful to understand leadership as one, albeit key, aspect of group dynamics more generally. Of course, decades-old intuitions within both behavioral political science and social psychology suggest that group membership causally shapes not only the way in which individuals see the world generally, but also the way in which they come to understand their own interests specifically in the course of complicated social and political interactions. Within political science, partisan affiliation is arguably the most heavily-studied kind of group membership. The classic formulation of Campbell et al (1960) that party identification raises a perceptual screen through which the individual tends to see what is favorable to his partisan orientation is suggestive that group membership has a causal effect because it influences cognition, and ultimately individual beliefs, about the political world. This line of argument has continued to the present day; contemporary research, employing more rigorous methodological techniques, suggests that individual beliefs concerning even basic factual information about politics and the social world can be strongly polarized by partisanship (e.g. Bartels 2002, Achen and Bartels 2006). The causal pathways that may give rise to such relationships are poorly understood; however, it is natural to suppose that an important role may be played by communications between group elites and group members. Such communications, of course, may take on a variety of different forms and 1

3 fulfill a number of different roles. Certain aspects of political leadership have been well-captured by game-theoretic models in which expert leaders have private information about the state of the world, and can sometimes strategically reveal or dissemble about this information in the context of a signalling game with fully-rational voters (e.g., Canes-Wrone, Herron, and Shotts 2001). At the same time, classical signalling models seem less than ideally equipped to confront the kinds of striking polarization in factual beliefs by partisanship detailed in the literature. While deception can (within limits) be sustainable in signalling-game equilibria, and while correlations between partisan affiliation and exposure to specific media outlets also doubtless play a role, the apparently robust nature of such behavioral phenomena cry out for further theoretical explication. This paper takes as its starting point the centrality of coordination problems to group life (Schelling 1960). Within the political arena, elections, wars, and other kinds of conflicts are often contests between competing partisan, national, ethnic, or other groups which must be able to engage in meaningfully coordinated action in order to meet their objectives. Previous research within a number of disciplines has argued that leadership can be one instrumentally useful means by which collectives can achieve coordinated outcomes more efficiently, from game-theoretic (Calvert 1992), lab-experimental (Wilson and Rhodes 1997), and evolutionary (Van Vugt 2006) viewpoints. This paper presents an analytic framework explicitly modelling one process, novel in the literature, by which group leaders might be able to induce coordinated action among their followers. A gametheoretic framework is developed, in which a leader with private information about the state of the world sends a message about it to her followers. In the model, the leader is best off when followers successfully coordinate their actions. Because followers have preferences over coordination outcomes that are aligned in some states of the world, but not in others, leaders sometimes have incentives to misrepresent the state of the world in order to make coordination more likely. Two novel refinements, Leadership-Correlated Equilibrium and Belief-Based Followership Equilibrium, explicate distinct mechanisms through which a leader s messages about the world could potentially coordinate the actions of fully-rational followers. In the first approach, Leadership-Correlated 2

4 Equilibrium, Followers use messages from a Leader as simple correlation devices guiding their actions. Under this approach, the way in which messages affect Followers beliefs, or the content of messages as factual statements about the world more generally, are effectively decoupled from Followers choices of actions. In the second approach, Belief-Based Followership Equilibrium, Followers use messages from a Leader as potentially informative signals to be used in updating their beliefs about the state of the world. Under this approach, Leaders messages can affect Followers prospects for coordination by shaping Followers perceptions of the underlying nature of their interaction with one another, which depend on their beliefs about the state of the world. The intuitions behind these refinements are then tested in a laboratory experiment. Results from the experimental games suggest that leaders do frequently misrepresent the state of the world to followers when it is in their interests to do so; that leaders strategically-chosen messages about the state of the world are highly effective in coordinating group members actions; and that leaders messages strongly influence followers beliefs about the state of the world even when Bayesian followers would not find the messages to be credible as statements of fact. Followers appear not fully to account for leaders strategic incentives to misrepresent the state of the world in forming their posterior beliefs, even though the experimental elicitation mechanism offers substantial monetary rewards for accuracy. This finding suggests behavioral foundations for the empirical regularity that members of different social groups may have different factual beliefs about the world. The paper contains five further sections. Section 2 presents a formal framework for analyzing leadership and coordination. Section 3 motivates and offers definitions for Leadership-Correlated Equilibrium and Belief-Based Followership Equilibrium, as well as presenting some benchmark theoretical results. Section 4 describes the experimental design as well as the conditions under which the experiments were carried out. Section 5 presents and discusses results from the experiment. Finally, Section 6 offers further discussion, sets the paper in a broader context, and concludes. 3

5 2 A Framework for Analyzing Leadership and Coordination This section describes a Leadership and Coordination game, in which a Leader sends a message about the state of the world to two Followers, who subsequently choose alternatives in a coordination game. 1 The ultimate payoffs of all actors Leader and Followers alike are determined by Followers choices. 2 In the game, the state of the world ω Ω={1, 2, 3} is determined by a random draw from a probability distribution for which ω = n with probability ρ n n {1, 2, 3}, where n ρ n = 1. The prior probabilities ρ n are common knowledge. In addition, the Leader knows the state of the world ω; Followers only know the prior probabilities. First, the Leader sends a message m {1, 2, 3} about the state of the world to both group members. This message is costless, and its contents may or may not correspond to the true state of the world. Then, each of two Followers, F A and F B, must choose either of two options, A or B. Because the interaction between these Followers takes on the form of a coordination game, each Follower prefers to choose A (B) given that her opponent will choose A (B). However, the precise nature of the payoffs depends on the state of the world, which from the perspective of Followers is ex ante uncertain. Followers F A and F B have preferences over coordination outcomes that are aligned in some states of the world, but in conflict in other states of the world. Specifically, as summarized in the payoff matrices below, for F A a coordination outcome (A, A) yields a payoff π = 1 when ω {1, 2} and a payoff π = μ (0, 1) when ω = 3; an outcome (B,B) instead yields a payoff π = μ when ω {1, 2} and a payoff π = 1 when ω = 3. In contrast, for F B, a coordination outcome (A, A) yields a payoff 1 For clarity of prose, throughout the Leader is referred to using female pronouns, while the Followers are referred to using male pronouns. 2 This analytic framework, involving a leader s attempts to influence the dynamics of a two-person group, is of course highly stylized; however, the underlying intuitions about the role of leadership in settings that place a premium on group coordination are meant to extend to interactions within larger groups as well. However, the literal-minded reader can also interpret the model more strictly, in terms of a group leader s attempts to affect social interactions within pairs of group members who may come to interact during the course of group life. In either case, it is easy to imagine that both leaders as well as group members may, in many situations, have an interest in successfully coordinated action even in the presence of within-group, across-individual differences as to which coordination outcome ought to serve as a locus of group action. 4

6 π = 1 when ω = 1 and a payoff π = μ when ω {2, 3}; an outcome (B,B) instead yields a payoff π = μ when ω = 1 and a payoff π = 1 when ω = {2, 3}. An outcome involving a coordination failure, i.e. (A, B) or(b,a), yields payoff π = 0 for both Followers in every state of the world. Payoff Matrix for State of the World ω =1.. F B : A F B : B F A : A 1, 1 0, 0 F A : B 0, 0 μ, μ Payoff Matrix for State of the World ω =2.. F B : A F B : B F A : A 1,μ 0, 0 F A : B 0, 0 μ, 1 Payoff Matrix for State of the World ω =3.. F B : A F B : B F A : A μ, μ 0, 0 F A : B 0, 0 1, 1 The Leader also has preferences over outcomes that depend on the Leader s type, θ {L A,L B }.A leader of type L A receives the same payoffs for every outcome, in every state of the world, as Follower F A ; in the same way, a leader of type L B receives the same payoffs from outcomes as Follower F B. The Leader knows her own type, while Followers share a common prior belief that θ = L A with probability q [0, 1] but that θ = L B with complementary probability 1 q. This preference structure reflects an intuitive account of group life. Successful coordination is essential for achieving many group ends; yet, members of a group may sometimes agree, and sometimes disagree, about the coordinated outcome they would most like to achieve. In the framework here, the two Followers agree about which coordination outcome they individually prefer in two of the three states of the world (ω = {1, 3}); these will be referred to as Agreement States. In the third state of the world (ω = 2), however, there is disagreement; ω = 2 will be referred to as the Disagreement State. The potential for disagreement alongside the possibility of agreement, all within the context of an intertwined destiny as modeled here by a coordination game, is typical of the interactions of political or other social groups. As an example, members of a given political party may share an interest in, 5

7 and need for, coordination for the purposes of winning power. However, different group members may exhibit differing tendencies; for example, some members of a given political party may be more inclined to be foreign policy doves, while others possess more hawkish inclinations. Such distinctions indicate the potential for disagreement under certain circumstances while the distinct factions may potentially find themselves in complete agreement under others. Hawks may favor bold intervention in some states of the world for which doves would find such action imprudent; however, in a state of the world following a direct attack against the country by a menacing threat, or in a state of the world where any present threat is considered by all sides to be minor and more amenable to negotiations, group members may be united in their preferences over courses of action. 3 Two Fully-Rational Conceptions of Leadership in Equilibrium This section describes two distinct, and complementary, approaches to modeling behavior using Perfect Bayesian Equilibria of the Leadership and Coordination Game. In the first approach, Leadership- Correlated Equilibrium, Followers use messages from a Leader as simple correlation devices guiding their actions. Under this approach, the way in which messages affect Followers beliefs, or the content of messages as factual statements about the world more generally, are effectively decoupled from Followers choices of actions. In the second approach, Belief-Based Followership Equilibrium, Followers use messages from a Leader as potentially informative signals to be used in updating their beliefs about the state of the world. Under this approach, Leaders messages can affect Followers prospects for coordination by shaping Followers perceptions of the underlying nature of their interaction with one another, which depend on their beliefs about the state of the world. Both Leadership-Correlated Equilibrium and Belief-Based Followership Equilibrium can be thought of as particular refinements of Perfect Bayesian Equilibrium in the context of the Leadership and Coordination game. In any state of the world, there are three equilibria in the coordination subgame following receipt of the Leader s message: coordination on (A, A), coordination on (B, B), and a Mixed-Strategy Nash Equilibrium. Because of this, there are a very large number of Perfect Bayesian 6

8 Equilibria in the Leadership and Coordination game as a whole. As such, the Leadership-Correlated and Belief-Based Followership Equilibrium concepts are far from exhaustive in characterizing potential equilibria in the game as a whole. However, these equilibrium concepts are useful in explicating some of the mechanisms through which leadership can potentially help group members overcome coordination problems of different kinds. In addition, both equilibrium concepts are supported by intuitive, behaviorally plausible refinement assumptions, which is not the case for some of the other Perfect Bayesian equilibria of the Leadership and Coordination game. Before proceeding to these equilibrium concepts, it is useful to define several terms that will enhance clarity of discussion. Definition 1. Self-Interested Agreement State. Suppose that in state of the world ω, a Leader s most-preferred outcome is X. Then the Leader s self-interested Agreement State ω is the state in which X is both Followers most-preferred outcome. According to this definition, a Leader of type L A has self-interested Agreement State ω = 1 when the true state is ω = {1, 2}, but ω = 3 when ω = 3; a Leader of type L B has self-interested Agreement State ω = 1 when ω = 1, but ω = 3 when ω = {2, 3}. Intuitively, Leaders can potentially benefit when Followers believe the state of the world to be the Leader s self-interested Agreement State, ω, because both Followers will perceive a preference for coordination on the Leader s most-preferred outcome. Definition 2. Self-Interested Message. A Leader s message m is a self-interested message if m = ω. Thus, a Leader s message will be referred to as self-interested if the message communicates to Followers that the Leader s self-interested Agreement State is in fact the state of the world. Note that, according to this definition, it is possible for a self-interested message either to be truthful (when ω = ω ) or untruthful (when ω ω ). 7

9 Leadership-Correlated Equilibrium One pathway through which Leaders communications may coordinate Followers is simply through providing focal points around which Followers may rally. In certain settings, in the absence of a message from a group Leader, neither A nor B may constitute a focal alternative for Followers. When this is the case, it seems natural to suppose that the mixed-strategy equilibrium of the coordination subgame will be selected. A Leader s message to her Followers may allow for the selection of a more efficient equilibrium, in which coordination failures never occur, by giving a focal property either to a choice of A or to a choice of B. In the framework presented above, Leaders send messages m whose manifest content is about the state of the world, ω. The preceding paragraph suggests however that such messages may influence Followers prospects for coordination in a way that is effectively decoupled from actors actual beliefs about the state of the world. A message m = 1, for example, may bestow a focal property on alternative A, because the coordination outcome (A, A) is Pareto-optimal in state of the world ω = 1. Potentially, such a message could serve as a purely rhetorical device that aids coordination independent of actors beliefs about the state of the world. Of course, in many circumstances, fully-rational Followers can nonetheless learn something about the state of the world from Leaders messages, even when such messages merely serve as coordination devices. Within the context of any separating equilibrium, different Leader types can be expected to send different messages in at least some states of the world. However, although Followers in such a context could use Bayes Rule to learn more about the state of the world, such updating of beliefs will not be directly relevant to Followers behavior in the Leadership-Correlated Equilibrium model; Followers will simply choose whatever alternative is made focal by Leaders messages. In the abstract, it is naturally the case that any number of assumptions could be made about which alternatives are made focal by which messages about the world. However, it seems behaviorally reasonable to make a refining assumption that focality is aligned with the natural-language meaning of messages received, in the following sense. In a context in which Followers actions cue off of Leaders 8

10 speech, a message m =1,evokingω = 1, would naturally call to mind the action A, because outcome (A, A) is Pareto-optimal under ω = 1. The definition of Leadership-Correlated Equilibrium includes this refining assumption: Definition 3. A Leadership-Correlated Equilibrium is a Perfect Bayesian Equilibrium of the Leadership and Coordination game in which: (i) Followers who receive an Agreement State message choose the alternative on which coordination would be Pareto-optimal in that Agreement State, that is, A for m = 1 and B for m =3; (ii) Followers who receive a Disagreement State message play the Mixed Strategy Nash Equilibrium of the coordination subgame because neither A nor B is focal; and (iii) Leaders choose their optimal message, given this profile of Follower responses. Given this definition, it is straightforward to establish the following Proposition: Proposition 1. The Leadership and Coordination game has a unique Leadership-Correlated Equilibrium, in which (1) the Leader sends her self-interested message and (2) Followers coordinate on the leader s preferred outcome. Sketch of Proof. By the definition of a Leadership-Correlated Equilibrium, both Followers choose A upon receiving m =1;B upon receiving m = 3; and both play the Mixed Strategy Nash Equilibrium upon receiving m = 2. Therefore, any Leader can obtain her most-preferred outcome by choosing her self-interested message. Leaders have no incentive to deviate from this strategy profile because they always receive their most-preferred outcome; an individual Follower has no incentive to deviate from this strategy profile because by construction the strategy profile is Nash in every coordination subgame. Thus, Leaders messages about the state of the world have the potential to organize Followers behavior even when Followers beliefs about the world are decoupled from their actions. From a technological standpoint, this coordination mechanism requires only that Agreement State messages can bestow a focal property on one particular alternative, an assumption that seems reasonable in many behavioral contexts. Nonetheless, the Leadership-Correlated Equilibrium model is quite thin as 9

11 a model of political communication. In many contexts, it seems natural to suppose that Followers beliefs about the world are not irrelevant to their behavior in equilibrium. To develop this intuition, the next section describes a second equilibrium concept for the Leadership and Coordination game. Belief-Based Followership Equilibrium In the Belief-Based Followership model, Followers beliefs about the state of the world do have relevance to Followers choices, and the ultimate prospects for successful coordination. Under this equilibrium concept, Leaders messages influence Follower behavior through a more subtle mechanism; rather than directly cueing Followers to choose one particular alternative over another, Leaders messages affect the beliefs of fully-rational Followers, and through this the Followers perceptions of their own interests, thereby affecting the shape of Followers coordination game interactions with one another. Consider the strategy profile in which, as in the previous section, a Leader always sends her selfinterested message. That is, a Leader of type L A sends m = 1 when ω = {1, 2}, but m = 3 when ω = 3, while a Leader of type L B sends m = 1 when ω = 1, but m = 3 when ω = {2, 3}. Given this strategy profile for the different Leader types, and prior beliefs about the state of the world defined by the ρ i, a Bayesian Follower will form posterior beliefs ρ i (m = j) about the state of the world given a message m = j {1, 3}: ρ 1 (m =1)=prob(ω =1 m =1)= ρ 1 ρ 1 +ρ 2 q ρ 2 (m =1)=prob(ω =2 m =1)= ρ 3 (m =1)=prob(ω =3 m =1)=0 ρ 1 (m =3)=prob(ω =1 m =3)=0 ρ 2 (m =3)=prob(ω =2 m =3)= ρ 3 (m =3)=prob(ω =3 m =3)= ρ 2q ρ 1 +ρ 2 q ρ 2(1 q) ρ 3 +ρ 2 (1 q) ρ 3 ρ 3 +ρ 2 (1 q) These posterior beliefs will be common knowledge for both Followers, given the structure of the game, since it is commonly known that both have access to the same information. Followers perceived preferences over the set of coordination outcomes, {(A, A), (B,B)}, will of 10

12 course be a function of their beliefs about the state of the world, since Follower payoffs are statedependent. For concreteness, suppose that in a given equilibrium context, Leaders always send their self-interested messages, as above. If Follower F B receives message m = 1, and has posterior belief ρ 1 > 1 2, then in expectation he believes that (A, A) would be his most-preferred outcome, rather than (B,B). Importantly, Followers posterior beliefs about the state of the world affect not only their individual beliefs about their privately most-preferred outcome, but also the kind of coordination game that both Followers commonly perceive one another to be playing. For instance, if Followers receive message m = 1, and share posterior belief ρ 1 > 1 2, then in expectation both Followers believe themselves to be best off when the outcome is (A, A). That is, the coordination game that the Followers perceive themselves to be playing is one with a Pareto-dominant Nash equilibrium. On the other hand, if Followers receive message m = 1, but share posterior belief ρ 2 > 1 2, then in expectation Follower F A believes himself to be best off when the outcome is (A, A), while Follower F B believes himself to be best off when the outcome is (B,B), because as described above, Followers preferences differ when ω = 2. That is, the coordination game that the Followers perceive themselves to be playing is a Battle-of-the-Sexes game, without a Pareto-dominant Nash equilibrium. While Followers payoffs are of course determined by the payoff matrix corresponding to the true state of the world ω, Followers behavior in the coordination game can naturally be expected to depend on the game they perceive themselves to be playing, given their posterior beliefs about the state of the world. Because some coordination problems can reasonably be thought of as more-easily solvable than others, and because the nature of the coordination game that is perceived depends on Followers posterior beliefs about the world, it is easy to see in this framework that Leaders with an interest in coordinating Followers actions will sometimes have an incentive to try to influence Followers beliefs in one or another direction, to the extent this can be done in equilibrium, consistent with Bayes Rule. That some coordination problems are more easily solved than others is reflected in the following 11

13 assumptions about the outcomes of play in the different coordination games that players perceive. 3 If, given their shared posterior belief about the state of the world, both Followers believe in expectation that the same outcome is best for each of them, then it will be assumed that both Followers coordinate on that outcome with probability 1. In other words, a common belief that (A, A) isbestforboth Followers leads to the selection of the (A, A) equilibrium in the coordination subgame; similarly, a common belief that (B,B) is best for both Followers leads to the selection of the (B,B) equilibrium in the coordination subgame. This assumption is meant to reflect that this perceived coordination game is relatively easy to solve because the players perfectly aligned incentives produce a strongly focal desirable outcome. Now suppose that the two Followers perceive their preferences over the two coordination outcomes to differ. This will be the case if Follower F A perceives (A, A) to be the best outcome for himself at the same time that Follower F B perceives (B,B) to be the best outcome for himself. In such a case it is assumed that the group members will fail to coordinate with probability F (0, 1) that is, the outcome of play will be one of the un-coordinated outcomes, (A, B) or(b,a), with strictly positive probability. Alternatively, they may in fact coordinate successfully on (A, A) or(b, B), each with probability 1 F 2 (this equiprobability reflecting the symmetry of the perceived game form). The introduction of a nonzero probability of coordination failure when group members perceived interests differ is meant to reflect that coordination is harder to achieve when incentives are imperfectly aligned than when players are in complete agreement. For the present purposes of defining the equilibrium concept, the value F will be taken to be rate of coordination failure when Followers play the Mixed- Strategy Nash equilibrium. Of course, in specific real-world settings, group members may be able to coordinate more often than the rate implied by the MSNE, though still not all the time, if they have access to some other partially-effective correlating device. In such instances F can be modelled differently, as appropriate; as a marker for such potential generalizations, F is left general in the notation. 3 These assumptions will be further justified momentarily. 12

14 As the results rely on the assumption that some coordination problems are more easily solved than others, it is worthwhile to reflect a moment on why this assumption might be a reasonable one. The perceived payoff matrices when group members believe themselves to be in agreement are coordination games with a focal, Pareto-dominant equilibrium, whereas the perceived payoff matrices when group members believe themselves not to be in agreement are battle of the sexes games. The relative ease with which coordination can be achieved in any concrete instance will of course vary depending upon a variety of factors, including the degree to which players are free to communicate; however, the intuition that coordination will on average be more easily achieved in coordination games with a Pareto-dominant equilibrium than in battle of the sexes games seems persuasive over a wide variety of conditions. If players must make choices without communication, a coordination game with a Pareto-dominant equilibrium will have a clear focal point that battle of the sexes games will lack, and as such, all other things equal, coordination can be expected to be easier in the former case than in the latter. This might be stated in another way. Both games have two pure strategy Nash equilibria and one mixed-strategy Nash equilibrium. Because there is a good reason for players to focus on one particular pure strategy equilibrium in the coordination game with a Pareto-dominant equilibrium, but there is no good reason for players to focus on one particular pure strategy equilibrium in the battle of the sexes game, they are more likely to play the mixed-strategy equilibrium in the battle of the sexes game - and this equilibrium involves a positive probability of coordination failure. Under conditions of free communication, the basic intuition remains intact. When group members believe their interests to be aligned, it is natural to expect that actors will quickly agree on the Pareto-optimal equilibrium virtually all of the time. Meanwhile, the battle-of-the-sexes game would pose a potentially non-trivial bargaining problem even under conditions of full communication. The experimental literature on bargaining provides some insight here: One of the clearest experimental results, which also accords well with field data, is that a nonnegligible frequency of disagreements [bargaining failures] is a characteristic of bargaining in virtually all kinds of environments. (Roth 1995) This is the case even when it is clear that it is in all actors interests to forge a deal: While this would be unsurprising if it occurred only in situations 13

15 that presented the bargainers with no mutually profitable agreements, most of the evidence suggests that disagreements and costly delays are pervasive even when it is evident that there are gains to be had from agreement. (Roth 1995) Because players may or may not perceive the state of the world correctly, the coordination game that players understand themselves to be engaged in may or may not correspond to the true payoff matrix reflecting the actual state of the world. While the probabilities of group members possible coordination outcomes are determined based on the payoff matrix that they perceive to be relevant, their ultimate payoffs are derived from the true payoff matrix corresponding to the actual state of the world. As a clarifying example, suppose that the state of the world is ω = 2, but imagine that both Followers instead have posterior belief ρ 1 = 1. In this case, both Followers perceive a preference for (A, A) as the best-possible outcome. According to the coordination-game assumptions, this implies that (A, A) will be the outcome. Group members payoffs are then determined according to the payoff matrix corresponding to ω = 2 that is, F A receives 1 while F B receives mu. This discussion motivates the following definition: 4 Definition 4. A Belief-Based Followership Equilibrium is a Perfect Bayesian Equilibrium of the Leadership and Coordination game in which: (A,A); (B,B); (i) Followers with posterior beliefs ρ 1 > 1 2 and ρ 2 =1 ρ 1 play A, resulting in coordination on (ii) Followers with posterior beliefs ρ 3 > 1 2 and ρ 2 =1 ρ 3 play B, resulting in coordination on (iii) Followers with posterior beliefs ρ 2 > 1 2 and either ρ 1 =1 ρ 2 or ρ 3 =1 ρ 2 play the Mixed Strategy Nash Equilibrium of the coordination subgame because neither A nor B is focal; (iv) Upon receiving any off-the-equilibrium path message m k, Followers have degenerate posterior 4 The discussion of Belief-Based Followership Equilibrium is focused around the strategy profile in which Leaders of either type always send self-interested messages. As will be seen in a moment, this strategy profile does not always support an equilibrium of the game. Additionally, when the strategy profile does support an equilibrium of the game, this equilibrium is not necessarily unique. The next draft of the paper will tie up these loose ends, either by further refining the definition of BBFE to ensure uniqueness when there is existence, or by characterizing the set of equilibria under the current definition of BBFE. 14

16 belief ρ k =1;and (v) Leaders choose their optimal message, given this profile of Follower responses. It will prove useful later to define one additional term inspired by Definition 4: Definition 5. A message m =1iscredible as a statement of fact if ρ 1 > 1 2 and ρ 2 =1 ρ 1. Similarly, a message m =3iscredible as a statement of fact if ρ 3 > 1 2 and ρ 2 =1 ρ 3. Given these definitions, it is straightforward to establish the following Proposition: Proposition 2. Consider a strategy profile in which a Leader of either type always sends her self-interested message in every state of the world. Then: Case I. When ρ 1 ρ 1 +ρ 2 q > 1 2 and ρ 3 ρ 3 +ρ 2 (1 q) > 1 2, this strategy profile is supported in a Belief- Based Followership Equilibrium, all Leaders messages are credible as statements of fact, and Followers coordinate on the Leader s preferred alternative; Case II. When ρ 1 ρ 1 +ρ 2 q < 1 2 and ρ 3 ρ 3 +ρ 2 (1 q) < 1 2, this strategy profile is supported in a Belief-Based Followership Equilibrium, any Leaders messages m =1or m =3are not credible as statements of fact, and Followers play the Mixed Strategy Nash Equilibrium; Case III. Suppose ρ 1 ρ 1 +ρ 2 q < 1 2 and ρ 3 ρ 3 +ρ 2 (1 q) > 1 (1 F)(1+μ) 2. When μ> 2, this strategy profile is not supported in a Belief-Based Followership Equilibrium, because a Leader of either type would have an incentive to deviate from m =1to m =3when ω =1. However, when μ< (1 F)(1+μ) 2, this strategy profile is supported in a Belief-Based Followership Equilibrium; Case IV. Suppose ρ 3 ρ 3 +ρ 2 (1 q) < 1 2 and ρ 1 ρ 1 +ρ 2 q > 1 (1 F)(1+μ) 2. When μ> 2, this strategy profile is not supported in a Belief-Based Followership Equilibrium, because a Leader of either type would have an incentive to deviate from m =3to m =1when ω =3. However, when μ< (1 F)(1+μ) 2, this strategy profile is supported in a Belief-Based Followership Equilibrium. Proof Sketch. Given the strategy profile in which a Leader of either type always sends her selfinterested message in every state of the world, consider the incentives of Leader types L A and L B. ω =1. The strategy profile assigns both Leader types to send m =1. Whenρ 1 = ρ 1 ρ 1 +ρ 2 q > 1 2, the outcome is (A, A) by the definition of Belief-Based Followership Equilibrium. This yields the maximum 15

17 payoff1tobothl A and L B so neither has an incentive to deviate. When instead ρ 1 = ρ 1 ρ 1 +ρ 2 q < 1 2, then ρ 2 > 1 2 ; the outcome is (A, A) with probability 1 F 2 1 F,(B,B) with probability 2, and a coordination failure with probability F. Thus, the expected payoff to either Leader type is (1 F)(1+μ) 2. Deviation to m = 2 yields posterior belief ρ 2 = 1, by the off-the-equilibrium-path assumption in the definition of BBFE, which yields the same distribution of outcomes, so there is no incentive to deviate. Finally, deviation to m = 3 yields posterior belief ρ 3 = ρ 3 ρ 3 +ρ 2 (1 q).ifρ 3 = again leads to an unchanged expected payoff (1 F)(1+μ) 2, while if ρ 3 = ρ 3 ρ 3 +ρ 2 (1 q) < 1 2, deviating to m =3 ρ 3 ρ 3 +ρ 2 (1 q) > 1 2, the outcome is (B,B) and the payoff to either Leader type is μ. Thus, each of the Leader types has an incentive to deviate if and only if both ρ 1 ρ 1 +ρ 2 q < 1 2 and ρ 3 ρ 3 +ρ 2 (1 q) > 1 2 while μ> (1 F)(1+μ) 2. ω =3. By a symmetric argument to that for ω = 1, each of the Leader types has an incentive to deviate if and only if both ρ 3 ρ 3 +ρ 2 (1 q) < 1 2 and ρ 1 ρ 1 +ρ 2 q > 1 2 while μ> (1 F)(1+μ) 2. ω =2. Leader type L A (L B )has the same preferences over outcomes, and according to the strategy profile sends the same message, as when ω =1(ω = 3). Hence, the incentives for deviation by L A (L B )fromm =1(m = 3) when ω =2 are the same as the incentives for deviation by both Leader types when ω =1(ω =3). Taken together, these results, along with the coordination-subgame equilibrium selection assumptions in the definition of BBFE, prove the Proposition. * FIGURE 1 ABOUT HERE * Figure 1 offers a graphical depiction of the conditions on the ρ s for the individual Cases in the Proposition. The axes depict the values of ρ 1 and ρ 2 ; the value of ρ 3 is implicit because ρ 3 =1 ρ 1 ρ 2. With the help of the Figure, the intuition behind Proposition 2 is quite straightforward. In Case I, the prior probability ρ 2 of the Disagreement State ω = 2 is quite low relative to the prior probabilities of the individual Agreement States. Under these conditions, a Leader s self-interested message, indicating one or the other of the Agreement States, will seem credible as a statement of fact to Followers. As a result, Leaders are able to obtain their first-best outcome, even when the true state of the world is a Disagreement State. In contrast, the prior probability of the Disagreement State is quite high relative to the prior 16

18 probabilities of the individual Agreement States in Case II. When this is true, a Leader s self-interested message will not seem credible as a statement of fact to Followers, who in the context of the Leader s strategy profile will believe that the Disagreement State is most likely regardless of the message received. Because a Leader cannot improve her position by deviating to another message, it is an equilibrium for Leaders always to send their self-interested message; however, these messages are not persuasive as they are in Case I, and Followers play the Mixed Strategy Nash Equilibrium of the coordination subgame. Finally, it is not an equilibrium for a Leader always to send his or her self-interested message in Cases III or IV when μ is sufficiently high. Consider Case III, in which the prior probability ρ 1 of ω = 1 is low relative to the prior probabilities of both other states. In these circumstances, a Leader s self-interested message m = 1 would not be credible as a statement of fact, given the strategy profile; Followers would believe the Disagreement State to be more likely than ω = 1. However, the prior probability of the other Agreement State, ω = 3, is sufficiently higher that a Leader s self-interested message m =3would be credible as a statement of fact. Because of this, Leaders have an incentive to deviate from m =1tom = 3, in order to ensure coordination (even if it is not on their most-preferred outcome), so long as μ is sufficiently large. 4 Experimental Instantiation The above theoretical exposition has developed several key intuitions. Central among these are that communications from leaders can facilitate coordination among group members, and that leaders may sometimes have incentives to misrepresent the state of the world in these communications. If group members are fully rational, certain messages from leaders will be understood as credible, while others will not be. Whether a message s credibility affects its usefulness as a coordinating device is a point of distinction between the Leadership-Correlated and Belief-Based Followership equilibrium concepts. A laboratory experiment was conducted as a means of exploring the intuitions underlying these theoretical approaches. Five experimental sessions were conducted in a social science lab at a large 17

19 American university. The 96 subjects, each of whom took part in one session only, interacted anonymously via networked computers; the experiments were programmed and conducted with the software z-tree (Fischbacher 1999). Participants, almost all of whom were undergraduates from around the university, signed up via a web-based recruitment system that draws on a large, pre-existing pool of potential subjects. Subjects were not recruited from the author s courses. After giving informed consent according to standard human subjects protocols, subjects received written instructions that were subsequently read aloud in order to promote understanding and induce common knowledge of the experimental scenario. At the beginning of each session, subjects were randomly assigned to a group of three people, consisting of a randomly-assigned Group Speaker and two Group Members ; these role labels were thought to be more neutral than Leader and Follower, their analogues in the theoretical exposition. 5 Group and role assignments remained fixed over 15 periods of interaction. Each period consisted of one play of the Leadership and Coordination stage game, followed by a Bonus Question for Followers that served as an incentive-compatible mechanism for eliciting their beliefs about the state of the world. The game-theoretic structure of the Leadership and Coordination stage game used in the lab was identical to that of the framework in Section 2. Subjects earned tokens, convertible into dollars at the end of the experiment (at a rate of 15 tokens = US$1), according to the outcomes of play (and, for Followers, their private Bonus Question responses); overall payoffs were equal to the sum of payoffs from each of the 15 periods, plus a US$5 show-up fee. The stage games in each of the 15 periods were formally independent that is, all random-variable quantities such as the state of the world and Leaders preferences were re-drawn independently in every period, in a way to be described in more detail. In addition, a variety of design features were intended to minimize the extent to which subjects could condition behavior on the previous period s outcome of 5 The Appendix contains a sample set of instructions to subjects, including an extensive series of screenshots showing the computer interface, that offers a complete depiction of the way the experiment was framed for participants. The description in this section places terminology from the experimental scenario in quotation marks where this differs from the theoretical exposition; for continuity, however, the analysis is presented in the same terms as the earlier discussion in the paper. 18

20 play. The coordination game alternatives (A and B in the theoretical exposition) were referred to using labels that varied from one period to the next; the elements in a given pair of alternatives were places, people, or things belonging to the same category (e.g., Cleveland/Cincinnati, Dandelion/Daffodil, Schooner/Sloop, etc.). It was intended that neither label in any pair seem particularly salient relative to the other, and each subject saw any given pair of labels only once. To further inhibit conditioning on previous periods, the states of the world (ω = {1, 2, 3} in the theoretical exposition) were referred to using the names of colors as labels; for every group, in every period, three color names were drawn at random for this purpose from a set of ten. 6 Various features of the on-screen interface were also randomized; for example, it was known to subjects that alternatives and states of the world were both presented in an order that had been randomized for every subject. Finally, to ensure that neither Follower s preferences became focal relative to the other s, each Follower was referred to on his own screen as you, while his counterpart Follower was referred to as your counterpart. On Leaders screens, the two Followers were referred to as Group Member 1 and Group Member 2 for clarity, but Followers never observed these labels, and it was known to all subjects that these labels were randomly reassigned from one period to the next. Subjects were aware that each group s state of the world was in fact randomly drawn based on the commonly-known likelihoods (prior probabilities) shown to the group. These likelihoods (ρ 1, ρ 2, and ρ 3,withρ 1 + ρ 2 + ρ 3 = 1) took on positive integer percentage point values (e.g., 9%, 53%, etc.) that were themselves randomly and independently drawn for every group, in every period, according to a procedure that was not described to subjects. This procedure modestly oversampled the region of the probability simplex corresponding to Case II (using one round of rejection sampling; 27.3% of the likelihood triples drawn came from this region, which covers 16.7% of the probability simplex) in order to increase Followers exposure to circumstances in which Leaders had clear incentives to misrepresent the state of the world. Aside from this oversampling (and rounding considerations), the likelihood triples were uniform draws from the probability simplex. 6 The colors were White, Gray, Black, Yellow, Orange, Red, Brown, Green, Blue, and Purple. 19

21 All five sessions were run with q =0.5; each Leader s type was independently drawn in every period, so the Leader was equally likely to share each of her Followers preferences in every period. Two sessions (with 21 subjects each) were run with payoff parameter μ =0.8, while two sessions (with 21, 21, and 12 subjects respectively) were run with μ = 0.2; because patterns of Leader and Follower behavior were highly similar across these two conditions, all analyses below pool data from all five sessions. For both Leaders and Followers, the payoff for coordination on one s most-preferred alternative was 10 tokens (= US$0.667), while a failure to coordinate yielded no payoff. In each period, Followers answered a Bonus Question after choosing a coordination game alternative, but before receiving any feedback about the outcome of play. This Bonus Question simply asked Followers to guess what the state of the world for the period actually had been; a correct answer earned 10 additional tokens, while an incorrect answer earned none. Thus, the Bonus Question served as an incentive-compatible mechanism for eliciting Followers posterior beliefs about the most likely state of the world. Because the 10 token (= US$0.667) reward for a correct answer was equal in magnitude to the maximum possible payoff from the play of the game, subjects were well-motivated by lab-experimental standards to learn the state of the world as well as possible. 7 It was common knowledge that Leaders payoffs were not affected by Followers Bonus Question answers; in place of Bonus Question payoffs, Leaders simply received a flat payment of 5 extra tokens per period. As feedback at the end of each period, Leaders and Followers were informed which alternatives had been chosen by the two Followers. Leaders, who of course knew the state of the world, therefore learned their precise payoffs for the period. Followers, who were at no point told the state of the world, did not learn their precise payoffs, either for game play or for their Bonus Question response. They 7 Typical psychology studies exhibiting biases in decision making do not offer subjects concrete incentives based on their responses; as a result, such studies are treated skeptically by many political scientists and economists. Any demonstration that some kind of biased behavior remains intact even when subjects responses are rewarded with a financial incentive makes a stronger case for that bias as a robust phenomenon. Of course, it is reasonable to imagine that the presence of such motivation leads subjects to attend to the experimental task differently than they would in the absence of the motivation, which may pose a problem for inference or for an experiment s external validity in some contexts. However, foreshadowing the results, this feature of the design actually strengthens inference in the present experiment; subjects exhibit a particular kind of bias in the lab even though they have additional incentives to be unbiased, relative to the incentives likely faced in real-world contexts. 20